Four Lectures on Real H^P Spaces

This book introduces the real variable theory of H^P spaces briefly and concentrates on its applications to various aspects of analysis fields. It consists of four chapters. Chapter 1 introduces the basic theory of Fefferman–Stein on real H^P spaces. Chapter 2 describes the atomic decomposition theory and the molecular decomposition theory of real H^P spaces. In addition, the dual spaces of real H^P spaces, the interpolation of operators in H^P spaces, and the interpolation of H^P spaces are also discussed in Chapter 2. The properties of several basic operators in H^P spaces are discussed in Chapter 3 in detail. Among them, some basic results are contributed by Chinese mathematicians, such as the decomposition theory of weak H^P spaces and its applications to the study on the sharpness of singular integrals, a new method to deal with the elliptic Riesz means in H^P spaces, and the transference theorem of H^P-multipliers etc. The last chapter is devoted to applications of real H^P spaces to approximation theory.

• Real Variable Theory of HP (Rⁿ)
• Spaces:
  • Definition of H^P (Rⁿ) Spaces
  • Non-Tangential Maximal Functions
  • Grand Maximal Functions
• Decomposition Structure Theory of HP (Rⁿ) Spaces:
  • Atom
  • Dual Spaces of H¹ (Rⁿ)
  • Atom Decomposition
  • Dual Space of H^P (Rⁿ)
  • Interpolation of Operators
  • Interpolations of H^P Spaces
  • Weak H^P Spaces
  • Molecule
  • Molecule Decomposition
  • Applications to the Boundedness of Operators
• Applications to Fourier Analysis:
  • Fourier Transform
  • Fourier Multiplier
  • Riesz Potential Operators
  • Singular Integral Operators
  • Bochner-Riesz Means
  • Transference Theorems of H^P-Multipliers
• Applications to Approximation Theory:
  • K Functional
  • H^P-Multiplier and Jackson-Type Inequality
  • H^P-Multiplier and Bernstein Type Inequality
  • Approximation by Bochner-Riesz Means at Critical Index